The behavior, adaptability, and survival of organisms, including pathogens or immune system components, depend critically upon the capability to formulate appropriate responses, at different time scales, to fluctuating temporal patterns of environmental cues such as ligand concentrations or stresses. Traditionally, mathematical studies have focused on constant environments and steady-state behaviors, not on naturally occurring signals and transient behavior. The ultimate goal of this work is to investigate, both theoretically and experimentally, the characteristics of biological responses to time-varying signaling and what information is encoded in them. The primary focus will be on the study of invariance with respect to sensory symmetries, with an initial emphasis on fold-change-detection (FCD), a property that confers signaling systems robustness to scale uncertainty. Recent molecular cell biology studies have experimentally discovered FCD behavior in prokaryotic chemotaxis pathways, allowing responses to broad ranges of chemoattractant concentrations, and in the eukaryotic epidermal growth factor ERK and the Wnt signaling pathways, allowing robustness of behavior in the face of large variations in protein abundances. One aim of this work is to develop new, and to significantly expand existing, theory of FCD and other symmetries. This aim will involve foundational work in areas of mathematics ranging from control theory to group representations. Broader significance will include new insights in nonlinear dynamics and systems biology, as the properties studied are emergent synthetic behaviors, not explainable easily from the properties of individual chemicals in isolation. A complementary aim is to develop an experimental platform to study transient behaviors, particularly in chemotaxis. New experimental tools will be established, based on microfluidic technology and on molecular level analysis of intracellular signaling (FRET), to afford unprecedented spatiotemporal control of stimuli during measurements and to allow the effective testing of mathematical predictions. Broader significance will be in fostering the development of new microfluidic technology of wide applicability in biology and biomedical engineering.